Central-limit theorem via sample size or sampling magnitude?

Question

I have a small application of the central limit theorem:

"Take a sample of 40 exponentially distributed random numbers and calculate their mean. Repeat 1000 times, record each measured mean and calculate the mean of those 1000 means."

Here's my Rcode:

nsim   = 1000
nsamp  = 40
lambda = 0.2

clt <- data.frame(m=numeric(nsim), s=numeric(nsim))

for(s in 1:nsim)
{
    clt$v[s] <- mean(rexp(nsamp, lambda))
    clt$s[s] <- sd(rexp(nsamp, lambda))
}   

cat(mean(clt$v), "\n")
    cat(mean(clt$s), "\n")

I can observe that with increasing number of simulations (i.e. increasing the repetitions of the sampling event), the mean of means and mean of variance more closely agree with the theoretical values (1/lambda).

What I wonder is, does the Central-limit theorem only depend on the number of sampling events (1000) or is it just as much dependent on sufficiently large sample size (i.e. 40 in this example)?

Answer 1

The CLT is generally considered a theoretical distribution. We don't actually generate 1000 (or 10,000 or 100,000) samples from it. Rather, we imagine what would happen if we were to replicate this experiment 1000 (or, technically, infinite) times.

The R code you post caps it at 1000, probably because by 1000 samples it will demonstrate the properties of the CLT. But, again, the CLT really says what would happen if you repeated it an infinite number of times.

So, to your question:

does the Central-limit theorem only depend on the number of sampling events (1000) or is it just as much dependent on sufficiently large sample size (i.e. 40 in this example)?

It depends on both (kind of). The number of sampling events actually is infinite (though the CLT properties hold with smaller distributions), and as the sample size increases, the more normal (and smaller the variance) the distribution of means becomes.

Answer 2

It does not matter for law of large numbers

Your application of the behaviour of the mean relates to the law of large numbers.

It is irrelevant how you create the average. If you average all your samples or first average groups does not matter, because the end result is the same.

Example

Say your sample is $1,2,3,4$ then you could group it and compute average of group averages, but the result is the same (if the group's sizes are equal).

$$\frac{(2+3)/2 + (4+1)/2}{2} = \frac{2+3+4+1}{4}$$

It matters for central limit theorem

The CLT relates to the distribution of a sample average. When you standardize this based on the population mean and deviation, then the final result approaches a Normal distribution under the right circumstances (independence, finite variance, etc.).

If your question would have been about the central limit theorem or about approximations with a normal distribution then the size of the samples matters (not the number of repetitions, although this may have an effect when you make a histogram).

There is already a question about this: Why does increasing the sample size of coin flips not improve the normal curve approximation?

It deals with the issue of the two different sizes (the sample size and the repetitions size), as well as with problems of drawing a histogram.

Answer 3

The central limit theorem states that you should take samples of sufficiently large size (40 in your example), but it does not provide a limit on the minimum number of such samples, i.e., the statement holds regardless of the number of samples. See point #1 in this answer.